How robustly can you predict the future?

Abstract

Hardin and Taylor [5] proved that any function on the reals-even a nowhere continuous one-can be correctly predicted, based solely on its past behavior, at almost every point in time. They showed in [6] that one could even arrange for the predictors to be robust with respect to simple time shifts, and asked whether they could be robust with respect to other, more complicated time distortions. This question was partially answered by Bajpai and Velleman [1], who provided upper and lower frontiers (in the subgroup lattice of Homeo + (R)) on how robust a predictor can possibly be. We improve both frontiers, some of which reduce ultimately to consequences of Hölder's Theorem (that every Archimedean group is abelian).If f : (−∞, t f ) → S is a member of R S, we could view P(f ) as an attempt to predict which member of S (which "state") the function f would/should/will pick out at "time" t f , if t f were in its domain. We gauge how well P makes predictions by asking: for which F ∈ R S and which t ∈ R does P correctly predict F (t), based solely on F | (−∞,t) ? I.e., for which F : R → S and t ∈ R does the equality (*)hold?If we restricted our attention to continuous F : R → S (with respect to some nice topology on S) then the problem would trivialize, since P F | (−∞,t) could simply pick out lim z t F (z), which depends only on F | (−∞,t) . But if |S| ≥ 2 then it is impossible to find a P such that the equation (*) holds for every function F : R → S and every t ∈ R. Simply fix any f : (−∞, 0) → S; since |S| ≥ 2 there is a (possibly non-continuous) function Hardin and Taylor [5] considered what happens when we require the equality (*) to merely hold for almost every t. We will say that an S-predictor P is good if, for all total

Sparse analytic systems

“…Let ∼ be the equivalence relation on R induced by the sparse analytic system via Lemma 6. The properties of ∼ listed in the conclusion of Lemma 6 satisfy the assumptions of Lemma 20 of Cox-Elpers [6], and that lemma tells us that if 𝑆 := R/∼ and P : R ⌣ 𝑆 → 𝑆 is any analytic-anonymous S-predictor,5 then P fails to predict the function 𝑥 ↦ → [𝑥] ∼ for almost every 𝑥 ∈ R. 6 In particular, there is no good analytic-anonymous S-predictor.…”

“…The notion of a sparse analytic system obviously generalizes to a sparse -system for any , and Lemma 6 easily generalizes to such systems. In fact, Section 4 of Bajpai-Velleman [2] and Section 5 of Cox-Elpers [6] can both be viewed as constructions, in ZFC alone, of sparse -systems (with ‘increasing bijections’ in [2] and ‘increasing smooth diffeomorphisms’ in [6]).…”

“…6 Strictly speaking, the statement of [6, Lemma 20] only implies that an analytic-anonymous predictor fails to predict on a positive -measure set. This is good enough to answer Question 3, since such a predictor would not be good.…”

“…This is good enough to answer Question 3, since such a predictor would not be good. But the proof of [6, Lemma 20] – which was due essentially to Bajpai-Velleman [2] – shows that an analytic-anonymous predictor can successfully predict only for those x lying in some fixed equivalence class, which, in the context of Lemma 6, is countable. So analytic-anonymous predictors fail to predict almost everywhere in this situation.…”